Beginning School Mathematics

A Guide to the Resource

Ministry of Education

Learning Media

Wellington

First published for the Ministry of Education by

Learning Media Ltd, Box 3293, Wellington, New Zealand.

© Crown copyright 1994

All rights reserved. Enquiries should be made to the publisher.

Dewey number 372.7

ISBN 0 478 05974 4

Item number 94/212

Learning Media wishes to thank Denise Whitehead and

Mary Geary for their help in writing this resource.

Beliefs About Mathematics Learning

Introduction

The Shape and Content of the New Zealand Curriculum

Beginning School Mathematics in Your Classroom

The Structure of Beginning School Mathematics

The Components of Beginning School Mathematics

The Spirit of Each Module

Assessment and Evaluation

Classroom Planning and Organisation

Looking at One BSM Activity

Enhancing Mathematics Learning

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Contents

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Beliefs About Mathematics Learning

The teaching of mathematics in New Zealand classrooms is based on the followingbeliefs.

• Children experience mathematical ideas, and develop mathematical skills andunderstandings, before they start school.

• Children learn by relating new ideas to their existing experiences and ideas.• Children learn best when ideas are presented in realistic and meaningful

contexts.• Effective mathematics teaching involves a range of approaches.• Children learn in different ways at different rates.• Concepts can be presented in a variety of ways to match different learning styles.• Learning involves taking risks in a supportive environment.• The learner must be at the heart of all planning.• Feedback can take many forms, and is an essential part of the learning process.• Critical reflection and effective communication of ideas are essential learning

tools.

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Introduction

Beginning School Mathematics is an activity-based resource designed to supportthe teaching and learning of mathematics in the early years. It comprises a widerange of materials, including teachers’ guides, booklets for pupils, card equipmentfor many of the activities, and overview charts that allow the teaching objectives tobe viewed in the wider context. A key feature of the design of BSM is the flexibilitywith which it can be used, allowing teachers to choose from a range of teachingapproaches, and to adapt the activities, or incorporate new ones, to suit the diverseneeds of children. The philosophy and content of this resource provide teacherswith valuable support in their implementation of Mathematics in the New Zealand

Curriculum.

BSM has been designed to support and empower teachers. The possibilities andpotential of this resource are as flexible as the teacher, who continues to hold thekey to its effective use.

— BSM In Your Classroom (introductory statement in Cycles 8-12)

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The Shape and Content of the New Zealand Curriculum

Two major documents provide a basis for the initiative to improve the teaching andlearning of mathematics in New Zealand schools.

The New Zealand Curriculum FrameworkThis document contains the official policy for teaching, learning, and assessment inNew Zealand schools. It provides a coherent framework for the more specificnational curriculum statements, which describe in detail the knowledge, skills,understandings, and attitudes from seven essential learning areas necessary for abroad and balanced education. Mathematics is one of these essential learningareas.

Mathematics in the New Zealand CurriculumThis national curriculum statement specifies achievement objectives at eight levels(through all the years of schooling) and provides suggested learning experiencesand examples of assessment activities for mathematics programmes in NewZealand schools. Throughout, there is an emphasis on providing learningexperiences which are meaningful to students and which lead to an understandingof the many applications of mathematics in the world beyond school.

The Beginning School Mathematics resource outlined in this booklet will supportteachers as they implement Mathematics in the New Zealand Curriculum in relation toThe New Zealand Curriculum Framework. BSM has been developed and trialled inconsultation with New Zealand teachers. The principles and content reflect theknowledge and experience of a wide range of educators in the light of the bestcurrent practice.

An important aspect of Mathematics in the New Zealand Curriculum is its focus onthe mathematical processes skills. These skills — problem solving, reasoning, andcommunicating mathematical ideas — are experienced through the concepts ofnumber, measurement, geometry, algebra, and statistics.

Problem Solving

Developing Logic and Reasoning

Communicating Mathematical Ideas

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The processes skills place an increased emphasis on children• posing questions for mathematical exploration• communicating the results of their mathematical explorations• using calculators and computers in correct and appropriate ways• exploring probability.

Relationships and Number

Geometry and M

easurement

Logic and Number

Math

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Number

Mea

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Geom

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Algebra

Statis

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Beginnn

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Mathem

atics in the New Zealand Curriculum

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Beginning School Mathematics in Your Classroom

The BSM resource has been designed to help teachers to respond to children’sneeds, while, at the same time, providing a base from which teachers can becomecompetent and confident in teaching mathematics. The aim of this resource is todevelop children who can• recognise, explore, and enjoy mathematics

in everyday situations• solve problems in a variety of ways• apply skills and transfer knowledge• interact and co-operate with others• take responsibility for their own learning• communicate their findings in many ways• feel good about themselves as successful learners

of mathematics.

The following key points embody the important features of the BSM resource.

Beginning School Mathematics

• is designed to match children’s developmental stages of learning

• encourages the use of varied grouping

• is language based

• uses a variety of materials

• uses a cyclic approach to developing ideas

• shows the progression and sequence of each concept

• has objectives that show clear lesson sequences

• develops mathematical knowledge, skills, and understandings

• is activity based

• allows for the maintenance of previous learning

• allows children to make choices

• enables flexibility when planning and meeting children’s needs

• includes assessment as an integral part of teaching and learning

• relates mathematics to the world in which the child lives

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The Structure of Beginning School Mathematics

The BSM resource is cyclic in design. Each successive cycle introduces anddevelops new learning objectives as well as reinforcing ideas and skills fromchildren’s previous work.

There are twelve cycles altogether. Each cycle is divided into three modules.The strands from Mathematics in the New Zealand Curriculum are covered withinthis structure.

In each cycle:• Module 1 focuses on number, algebra, and statistics• Module 2 focuses on geometry, measurement, and number• Module 3 focuses on number, algebra, geometry, logic, and reasoning.

The mathematical processes skills — problem solving, reasoning, andcommunicating mathematical ideas — are learned and assessed within the contextof the more specific knowledge and skills of number, measurement, geometry,algebra, and statistics.— from Mathematics in the New Zealand Curriculum, page 23

Each cycle is accompanied by a checkpoint booklet which can be used in a varietyof ways to assess individual children’s understanding of key ideas in the cycle.

Cycle 1

Module 1

Cycle 1

Module 2

Cycle 1

Module 3

ComparisonsandRelationships

Shape,Movement,and Position

Classification,Order, andPattern

BSM

Cycle 1

Three modules in each cycle

One checkpoint for each cycle

Checkpoint 1

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The Components of Beginning School Mathematics

The BSM resource has five major components.

Overview ChartsThere are two overview charts, one covering Cycles 1 to 6 and the other Cycles 7to 12. The charts are essential planning documents, showing clearly where an ideais introduced or where the development of an idea occurs. The objectives from thetwelve cycles are presented on the charts in a way that makes the development ofideas over time an easy process to follow. The format also emphasises how themathematical ideas link with the strands identified in Mathematics in the New

Zealand Curriculum.

The overview charts will• help teachers to understand the development of ideas in terms of the overall

resource• assist in planning and preparation• be a useful support at parent conferences.

Overviewcharts

Equipment —Card Equipmentand Hardware

Children’sBooklets

Teachers’Books Checkpoints

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allows teachers and parents to see where a childis working within the whole resource

illustrates theintegration ofmathematicalprocesses

provides linkswith thecurriculumstrands

provides astarting pointfor planning

Ideas connectbetween cycles,and betweenmodules 1 and3 within eachcycle.

shows theprogressionandsequencing ofeach concept,enabling ideasto be tracedback and forth

enables teachersto track thedevelopment ofan achievementobjective

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Teachers’ BooksThe key component of each module is a teachers’ book which contains aims,objectives, information on mathematical ideas, and a selection of numberedactivities. The introduction of the booklet outlines• the spirit of the module• the key content• notes for the teacher• notes on the mathematical ideas to be introduced.Note: In Cycles 1 to 6 there are separate teachers’ books for each module. InCycles 7 to 12, the teachers’ material for the three modules of each cycle is in asingle book.

The objectives for the module are presented in table form, and are linked with theirassociated activities.• Core activities, numbered 1 to 20, are designed to be used by the teacher

working with a group of children.• Class activities, numbered 21 to 40, provide the opportunity for the teacher to

supplement the core activities.• Independent activities, numbered 41 to 80, are designed to provide maintenance

for ideas covered previously.• Extension activities, numbered from 81 onwards, provide a challenge for

children who show a mastery of particular ideas.

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The numbering system is consistent throughout the resource, and allowscomponents to be easily identified for borrowing or reshelving. Although thenumbering system supports a structured use of the materials, it also allows forflexibility.

The emphasis throughout the resource should be on the objectives, and on helpingchildren to achieve them. The activities contain mathematical information, ideas,and suggestions for teachers. Teachers should feel at liberty to alter or adaptactivities, to supplement them, or to develop and collect other activities which areparticularly suitable for their children, community, or environment.— BSM In Your Classroom (introductory statement in Cycles 8-12)

9 - 3 - 51 They deserve a place

activity number

module num

ber

cycle number

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CheckpointsThe checkpoint booklets that accompany each cycle suggest two methods ofassessing children’s progress — day-to-day monitoring, and checking children’sunderstanding of key ideas. Through these two approaches, teachers can planappropriate ways to meet the assessment needs of children learning at variousrates. By using the checkpoint, a teacher can assess the child’s understanding ofsome of the ideas introduced in that cycle. For each checkpoint, relevant objectivesand necessary equipment are shown, and a range of questions is suggested. Thebooklet also contains day-to-day recording sheets for each module, showingobjectives with their associated activities, and providing space for teachers’anecdotal notes.

Children’s BookletsBooklets for the children’s own use are first introduced in Cycle 7. They aredesigned to extend the children’s knowledge and skills and to provide furtheropportunities to develop responsibility and independence in learning. In somecases, the games and activities require items of BSM card equipment, or otherstandard mathematics equipment, such as Cuisenaire rods or counters.

The booklets also provide children with a first “maths book” and give thempractice in following written and pictorial instructions.

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Equipment — Card Equipment and Hardware

Card EquipmentCard equipment is supplied to match particular activities in the resource. It isnumbered to match the activity.

Some of the card equipment, such as number cards and digit labels, are for useacross the twelve cycles.

Card equipment for a cycle may also include• copymasters and recording sheets for the children’s use• direction cards (for some activities) designed to be read, interpreted, and

followed by the children independently.

HardwareBSM hardware, and other standard mathematical equipment required for BSMactivities, is described in the teachers’ books. These materials are not availablefrom Learning Media, but can be obtained from local reputable suppliers ofeducation equipment.

Miro Kowhai Koromiko

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The Spirit of Each Module

The “Spirit of the Module” statements, found in the introduction to each teacher’sbook, summarise the intent of each module. These statements are useful whencross referencing the BSM resource with the achievement objectives in Mathematics

in the New Zealand Curriculum.

Children explore a wide rangeof natural and environmentalmaterials individually andinformally. These experiencescontinue the way in whichchildren gained earlymathematical ideas before theyentered school.

As the children match pairs ofrelated objects, numbergroups, lengths, and weights,they demonstrate an increasingunderstanding ofrelationships.

Children have experiencessimilar to those in which theyhave explored spatialrelationships and shapesbefore they entered school.They develop the relatedlanguage.

Children develop the idea ofcomparison and the ability tomake discoveries based on thisidea.

As children are physicallyinvolved in exploringsituations within a structuredsetting, they develop the ideasand language of shape,movement, and position.

As the children constructshapes, and move their bodiesand objects in differentdirections and to particularplaces, they extend theirspatial understanding.

Children explore theproperties of everyday objectsto make them aware oflikenesses, differences, andorder.

Children are helped to becomeaware of patterns, their varietyand structure. They begin toput their knowledge oflikenesses and differences touse.

As children discover that anobject has several attributes,they widen their bases forclassifying.

Cycle 1 2 3

Module

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odule

2M

odule

3

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As the children use theirenumeration skills, classify setsfor number property, andexplore number patternswithin sets, they begin tounderstand the constancy of anumber.

As children study eachnumber group in depth, theybring together anunderstanding of theconstancy of each number, itsposition in the series ofcounting numbers, and itsrelationship to other numbers.

As children compare numbergroups, they begin tounderstand relationshipsbetween numbers.

Children are involved co-operatively in finding answersto questions about position,distance, time, length, andweight.

Children’s spatialunderstanding is increasedthrough shape and movementactivities that involve directionand orientation, and alsothrough a variety of measuringtasks.

Children discover that there aremany three- and four-sidedshapes. They learn new waysof measuring weight andcapacity by using identicalobjects and containers.

As children begin to usepictorial symbols and stylisednumber patterns to label sets,their classification skillsdevelop further.

Children have experience insolving problems byrecognising the structures inthe materials used, makingchoices, and deciding whetherthese are appropriate.

Children’s ability to solveproblems is further developedby the teacher’s questions.The children suggest varioussolutions to problems anddemonstrate these withequipment.

4 5 6

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Cycle 7 8 9

Children’s understanding ofrelationships, the constancy ofnumbers, and the additionoperation is shown as theymake relationship charts andpicture graphs independently,and record names for anumber, and additionequations.

Children’s experience withnumbers is extended as theyjoin sets and describe theaddition operation.

As children are encouraged tobring their own ideas intomathematics and record intheir own ways, they begin tosee mathematics as a part ofeveryday life and tounderstand that facts remainthe same, regardless of howthey are recorded.

Children’s exploration ofspatial relationships is relatedclosely to measurement as theycover surfaces with shapes andmake judgements aboutdistances.

Children’s understanding ofmeasurement is demonstratedby their ability to estimate, andthen measure accurately withsuitable unconventional units.

Children continue to exploreunconventional measures,while using conventionalmeasures informally. They areencouraged to judge theappropriateness of particularmeasures.

Module

1M

odule

2M

odule

3

Children’s thinking isextended by furtherexperiences with inclusion andintersection in classificationactivities, and by the ideas ofopposite actions and reverseorder in number operations.

Children study a set to nameits attributes, includingnumber property, or to seepatterns of number groupswithin a set of a particularnumber.

Children are encouraged torecord in their own ways andexperiment with other ways.They should, at the same time,meet more formal types ofrecording. Some are beginningto see that conventional formsof recording are necessary forprecise and standardisedcommunication.

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10 11 12

The children continue to refinetheir thinking in mathematics.As they work at solvingproblems, they develop furtherstrong links between the use ofconcrete materials and formalrecording.

As children work with ideasintroduced in previousmodules, they have furtheropportunities to work in theabstract, as well as continuingto work with concretematerials.

As children work withnumbers in the higherdecades, they are encouragedto explain their thinking andrecording to other children,and to use a range of materialsto show the same idea.

Children are consolidatingconcepts of measurement oflength, weight, capacity, andtime as they use bothconventional andunconventional measures inthese areas.

As the children move, measure,and label, they realise that theyuse many ideas and skills tosolve a problem, and that someskills are more relevant thanothers.

As the children work withconventional measures oflength, distance, weightcapacity, time, and money,they widen their experience ofmathematics in everyday life.

The children work withnumber and operations,estimating the numbers inlarger sets, and looking forexceptions that could disprovea statement.

As the children work with theactivities in this module, theylink ideas from previousmodules to solve problems innew contexts. They are furtherencouraged to look for themathematics in everydaysituations.

As children work through thismodule, they continue to useconcrete material but begin torely more on their ability towork using only the symbols.

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Assessment and Evaluation

Assessment should focus on both what children know and can do, and on howthey think about mathematics.

Assessment should be integral to the normal teaching and learning programme....[It] should involve multiple techniques, including written, oral, and demonstrationformats. Group and team activities should be assessed.

It is expected that, in assessing students’ progress, teachers will make judgments asto an individual’s degree of achievement of particular objectives, and will includecommentary on that degree of achievement when reporting to parents.

— Mathematics in the New Zealand Curriculum, pages 15 and 16

Day-to-day RecordingAnecdotal and/or checklist recording, based on observation and informaldiscussion, can be used to assess the child’s achievement of the objectives. Byobserving and interacting, teachers can gain a great deal of information on eachchild’s progress. If this information is recorded, it can be used to pace children andto plan for further learning. Children will progress at their own rate, and ongoingobservation will help to meet their individual needs.

CheckpointsCheckpoints can be used in a variety of ways to check individual children’sunderstanding of some of the ideas in a cycle. Guidelines on purpose and use canbe found in each checkpoint.

Points to think about.• How will you collect data that demonstrates the child’s degree of

achievement of the objective?• Are you using a variety of assessment techniques?• What retainable evidence will you collect? How will you

collate this information?• What are the school’s policy requirements for assessment

and evaluation?

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Classroom Planning and Organisation

There are various ways to approach programme planning. Whichever approachyou adopt, be mindful of the following questions.• Is the learner at the heart of your planning?• Does your current planning reflect the requirements of Mathematics in the New

Zealand Curriculum?

• Does your current system for the storage of resources allow for easy access?• Have you considered a range of planning options? For example:

A balanced mathematical programme includes concept learning, developing andmaintaining skills, and learning to tackle applications. These should be taught insuch a way that students develop the ability to think mathematically.

— Mathematics in the New Zealand Curriculum, page 11

• Do I need to supplement my group box with other activities to maintainprevious learning?

• I know there is an appropriate class activity in 3.1 that supports this idea.

• During this unit, how can I gather data for assessment purposes?

• Will there be an opportunity during the day to develop this idea further?

• What would be the best way to groupthe children?

• What context will I use to make thispersonally meaningful to the children?

• I think the problem-solving process “useequipment appropriately when exploringmathematical ideas” would link in here.

• What could I use as a starter problem toget the children thinking mathematicallyand help me determine readiness?

• What will I focus on for assessment?

• If the children design a chart, should I keep it for assessment purposes?

• What BSM objectives and activities would help as we work towards this idea?

• This relates to our science unit on kites.

• What other resources do we have that could be used here?

• This idea fits in with the measurement topic from Module 2, and we will revisitit during Module 3.

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Looking at One BSM Activity

Mathematics in the New Zealand Curriculum

Measurement Strand, Levels 1 and 2

Achievement Objectives• Carry out practical measuring tasks, using appropriate metric units for length,

mass, and capacity.• Use equipment appropriately when exploring mathematical ideas.• Use their own and mathematical language to explain mathematical ideas.

Think of other ways that this objective could be developed.• Use a variety of containers to develop the concept of measurement.• Relate the objective to a real-life context where liquid is measured in litres.• Integrate the objective with a science unit which requires the measurement

of liquid.• Allow the children to think of different situations or problems to develop this

objective, for example, “How can we find out how much drink there is in ourgroup’s drink bottles?”

• Work across strands.

This activity from Cycle 8 Module 2:

• allows for the use of prior knowledge

• provides a relevant context

• lets the teacher explore the child’sconcept of a litre

• relates learning to the child’s ownexperiences and ideas

• allows children to explore conceptsand skills

• promotes active learning

• allows children to take risks

• enables the learner to reflect on actionsand experiences

• provides a situation where children canwork as a team to carry out amathematical exploration.

Objective Six – ActivityObjective6 Associate the term: litrewith capacity

6 A litre is a very big drink.MaterialsA large container that holds approximately 1 litre, e.g., a large preserving jar.

8 identical yoghurt containers.A litre of orange drink or milk.Activity (up to 8 chidren)Talk briefly about "all that drink" being 1 litre. Discuss the sorts of things

we buy in litres, e.g., petrol, oils, drinks, detergent.Use a real life situation to fit the next part of the activity.Ask the children to share out the drink into the yoghurt containersso that everyone gets a fair portion.Talk about each child having the same quantity.

Tip it all back and emphasise that altogether it would be a very big drink!

Share it out again and let the children drink their share.

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Enhancing Mathematics Learning

The greatest resource of all is the children themselves. These children bring a richtapestry of knowledge and experiences to the classroom. They have alreadyencountered and used mathematics in their own lives, within contexts that arepersonally meaningful. Our challenge, as teachers, is to provide an environmentthat presents realistic learning situations that allow children to continue weavingtheir tapestry.

The requirements of the New Zealand Curriculum Framework and Mathematics in the

New Zealand Curriculum, and the philosophy of the BSM resource, haveimplications for how mathematics learning can best be enhanced.

Classroom EnvironmentThe role of the teacher is to create a positive learning environment by:• engaging children in meaningful and purposeful tasks• modelling strategies for working with specific skills and problems• demonstrating a positive attitude to mathematics• providing time for thinking, reflecting, reacting, and sharing• expecting children to be able to:

solve problemswork co-operativelytake responsibilitytake risksreason logicallycommunicate mathematicallyestimate and check the reasonableness of answers.

ResourcesThere are various resources available to support the concepts being developedfrom Mathematics in the New Zealand Curriculum. Beginning School Mathematics isone resource and can be supplemented by others. Teachers need to evaluate anyresources used according to the needs of their pupils.

Parents and BSMTeachers need to communicate ways in which home and school can work togetherto support children as they learn mathematics. The leaflet Beginning School

Mathematics: A Pamphlet for Parents provides some starting points, and suggests arange of ways in which parents can encourage children in their developingunderstanding of mathematics.

Further support is available through Teacher Support/Advisory Services andbranches of the Family Maths Trust.

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The contribution of teachers, advisers, and children has been a major factor in thetrialling and development of BSM. Educators throughout New Zealand areexploring a variety of options for using the BSM resource to support Mathematics in

the New Zealand Curriculum. Beginning School Mathematics, alongside otherresources, is an invaluable support for mathematics teaching and learning in juniorclassrooms.